Limiting behaviour and modular completions of MacMahon-like q-series

TL;DR

This paper investigates properties and extensions of MacMahon-like q-series, revealing new arithmetic structures, modular completions, and their approximation of colored partition functions, thus deepening the understanding of their connections to modular forms.

Contribution

It introduces infinite families of MacMahon-like functions approximating colored partitions and eta quotients, and uncovers new modular and arithmetic structures.

Findings

Established infinite families approximating colored partition functions

Discovered new modular completions of MacMahon-like functions

Revealed suggestive arithmetic structures in these functions

Abstract

Recently, MacMahon's generalized sum-of-divisor functions were shown to link partitions, quasimodular forms, and q-multiple zeta values. In this paper, we explore many further properties and extensions of these. Firstly, we address a question of Ono by producing infinite families of MacMahon-like functions that approximate the colored partition functions (and indeed other eta quotients). We further explore the MacMahon-like functions and discover new and suggestive arithmetic structure and modular completions.